*Fractal Geometry*has a focus on different definitions of dimension, while

*Fractals Everywhere*makes iterated function systems central. The third book,

*A Tale of Two Fractals*, is more specialised, focusing on some specific questions about two particular fractals. With space taken up by proofs and exercises, the breadth of coverage of all these books is necessarily relatively constrained. One consequence is that they have surprisingly little overlap; they are in fact quite complementary and I worked through them partly in parallel.

Falconer's ** Fractal Geometry: Mathematical Foundations and Applications**
(third edition 2014, first edition 1997) is a fairly traditional text,
aimed at higher undergraduate or graduate mathematics students and
not so accessible to those from other disciplines. Falconer makes
some concessions: he keeps the measure theory to a minimum and states
the occasional theorem without giving a proof, while sections with
harder material are marked with asterisks and "may be omitted without
interrupting the development". And a few sections are almost entirely
descriptive, for example the brief look at the Lorentz attractor in
"Continuous Dynamical Systems". But otherwise the presentation is
rigorous, and clear but dense.

It will help readers if they have some familiarity with fractals already. An introductory chapter provides only a tiny bit of context, rapidly introducing classic fractals such as the Cantor set and the Sierpinski gasket, more for later reference than as motivation. After that Falconer gets straight down to the nitty-gritty.

Part one, "Foundations", begins with a chapter of assumed mathematics: metric spaces and suchlike, measures and mass distributions, a bit of probability. It then proceeds with chapters on box-counting dimension, Hausdorff and packing dimensions, methods for calculating dimensions, the extension to fractals of concepts such as density and tangents, and constraints on the dimensions of the projections, products and intersections of fractals. There's no attempt to be comprehensive here, but rather to introduce the key concepts, prove some major theorems and results, and give a feel for how different concepts of dimension work and are used.

Part two, "Applications and Examples", is broader in scope, but the applications are largely to other areas of mathematics and the examples mostly pull out individual interesting bits of mathematics. The topics covered include iterated function systems (with a glance at image compression), fractals in number theory (in digit distributions, and in continued fractions and Diophantine approximation), dimensions of graphs (self-affine functions and coastlines), Julia sets and the Mandelbrot set, and random fractals; a brief glance at dynamical systems does no more than touch on the logistic map. There is a final chapter at the end on "physical applications", but even this mostly highlights bits of associated mathematics: a continuous model for diffusion-limited aggregation, constraints on the dimension of singularity sets of potential functions, and single aspects of turbulence, antennas, and finance.

So *Fractal Geometry* is rewarding, but to those who can appreciate
mathematics for its own sake. There are exercises at the end of each
chapter, with full solutions available online.

Barnsley's ** Fractals Everywhere** (third edition 2012, first edition
1998) takes quite a different approach, building on the core concept of
an iterated function system and not introducing concepts of dimension
until nearly half-way through. The approach is formal and rigorous
(though a few proofs are omitted) and in many ways quite abstract,
but the coherent focus on iterated function systems and a plethora of
concrete examples, especially ones with two-dimensional visualisations,
provide good intuition and motivation.

Barnsley begins with basic metric spaces, then introduces the space of fractals (defined very broadly as the space of compact subsets of an underlying space, with the Hausdorff distance as a metric) and the key notion of an iterated function system (broadly, a complete metric space with a set of contraction mappings) and proves key Contraction Mapping and Collage theorems. He then introduces "code space" and symbolic addressing and dynamics on fractals, working up to the Shadowing Theorem showing that inaccurately calculated orbits are still useful. There's also a brief overview of different approaches to fractal dimension.

The second half of *Fractals Everywhere* applies and generalises
this toolkit. So the construction of fractal interpolation functions
is done using iterated function systems, Julia sets are presented as
the attractors of iterated function systems, measure theory allows a
formal definition of a fractal as a fixed point in a space of measures,
and the final chapter uses recurrent iterated function systems as a
tool for designing fractals. And there are other goodies, such as a
generalisation of the Mandelbrot set to the notion of a parameter space.

There's a good selection of exercises in *Fractals Everywhere*, cleverly
mixed up with examples, so one finds oneself attempting them almost
before realising it; there are also detailed solutions for most of them.
The black and white illustrations and diagrams, though relatively low
resolution, are effective. There are also thirty two pages of colour
plates, half of them supporting the main text, mostly illustrating
applications to art and image processing, and half forming a kind of
photo-essay on fractal art.

There are some suggestions for computer exploration, in a few places with
some (now old-fashioned BASIC) code. And the applications discussed are
mostly to image processing, with little on the physics and engineering
aspects of fractals (there is nothing on the Lorentz Attractor,
for example). So *Fractals Everywhere* may particularly appeal to
those with a computer science background. It's a nicely developed
exposition, however, covering some fascinating mathematics with verve,
so I recommend it to anyone with a background in formal mathematics who
is curious about fractals.

In ** A Tale of Two Fractals** Kirillov attempts to reach from fairly
elementary undergraduate mathematics — assuming only basic analysis,
linear algebra, geometry, group theory and so forth — to novel research
results and questions. He manages this by focusing on just two fractals,
the Sierpinksi and Apollonian gaskets, and a few topics on those: the
Laplace operator and harmonic functions on the Sierpinski gasket and
Descartes' Theorem and some geometry and group theory on the Apollonian
gaskets. So anyone after a general introduction to fractals should
look elsewhere.

The approach is fairly rapid and mostly rigorous, but makes no attempt to be systematic. There are a few "in-line" exercises and some more difficult problems. And there's a good selection of diagrams, some of them in colour. More than a quarter of the text is incorporated into "Info" sections providing background which is necessary but peripheral to the main discussion. These are labelled alphabetically, confusingly out of sequence with the rest of the (numerically ordered) material.

Kirillov does a decent job of selecting material with little in the way of prerequisites, but I found many places where things stated as "obvious" were not so clear, and there are places where the terminology or assumptions seem closer to graduate level. There are also a few significant typographical errors, notably in the diagrams, where the labelling often seems inconsistent with the text. Better proofing would have helped, perhaps with more feedback from undergraduate readers.

Despite these problems, *A Tale of Two Fractals* might appeal to hardy
undergraduates who have research ambitions already and want something
out of the syllabus to explore. It offers a good variety of material,
incorporating a mix of geometry, analysis, algebra, and number theory.

November 2016

**External links:**-
*Fractal Geometry*

- buy from Bookshop.org (UK)

- buy from Amazon.com or Amazon.co.uk

*Fractals Everywhere*

- buy from Bookshop.org (UK)

- buy from Amazon.com or Amazon.co.uk

*A Tale of Two Fractals*

- buy from Bookshop.org (UK)

- buy from Amazon.com or Amazon.co.uk

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**Related reviews:**-
- Peitgen et al. -
*Chaos and Fractals: New Frontiers of Science*

- Kenneth Falconer -*Fractals: A Very Short Introduction*

- Manfred Schroeder -*Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise*

- books about mathematics

- books published by Dover